Intbar

intbar in native mode $⨍ ⨎$

intbar in svg mode $\intbar \intBar$

Integrale

MathML

Mittelwertintegrale

Integral mit zwei Strichen: $\underset{A}{⨎} f (x) d x$
Integral mit einem Strich: $\underset{A}{⨍} f (x) d x$
Integral mit einem Strich in Textstyle: $\underset{A}{⨍} f (x) d x$
Integral mit einem Strich und mit Limits: $\underset{A}{⨎} f (x) d x$

Normale Integrale

Normales Integral über eine Menge mit displaystyle: $\int_{A} f (x) d x$
Normales Integral über ein Intervall mit Limits-Parameter mit displaystyle: $\int_{0}^{\infty} f (x) d x$
Normales Integral über ein Intervall (ohne Limits-Parameter) mit displaystyle: $\int_{0}^{\infty} f (x) d x$
Normales Integral über ein Intervall mit NoLimits-Parameter mit displaystyle $\int_{0}^{\infty} f (x) d x$
Normales Integral über ein Intervall (ohne Limits-Parameter) mit textstyle: $\int_{0}^{\infty} f (x) d x$
$\oint_{\partial Σ} ⟨ F, τ ⟩ d s = \iint_{Σ} ⟨ r o t F, ν ⟩ d S = \iint_{Σ} (\frac{\partial v_{2}}{\partial x} - \frac{\partial v_{1}}{\partial y}) d x d y$
$∭_{V_{K a r t}} f (x, y, z) d x d y d z = ∭_{V_{K u g}} \tilde{f} (r, θ, φ) \cdot r^{2} \sin θ d r d θ d φ$

Native

Mittelwertintegrale

intbar in native mode mit textstyle $⨍ ⨎$
intbar in native mode mit displaystyle $⨍ ⨎$
intbar in native mode mit displaystyle und limits $⨍_{0}^{\infty} ⨎$

Normale Integrale

Normales Integral über eine Menge mit displaystyle: $\int_{A} f (x) d x$
Normales Integral über ein Intervall mit Limits-Parameter mit displaystyle: $\int_{0}^{\infty} f (x) d x$
Normales Integral über ein Intervall (ohne Limits-Parameter) mit displaystyle: $\int_{0}^{\infty} f (x) d x$
Normales Integral über ein Intervall mit NoLimits-Parameter mit displaystyle $\int_{0}^{\infty} f (x) d x$
Normales Integral über ein Intervall (ohne Limits-Parameter) mit textstyle: $\int_{0}^{\infty} f (x) d x$
$\oint_{\partial Σ} ⟨ F, τ ⟩ d s = \iint_{Σ} ⟨ r o t F, ν ⟩ d S = \iint_{Σ} (\frac{\partial v_{2}}{\partial x} - \frac{\partial v_{1}}{\partial y}) d x d y$
$∭_{V_{K a r t}} f (x, y, z) d x d y d z = ∭_{V_{K u g}} \tilde{f} (r, θ, φ) \cdot r^{2} \sin θ d r d θ d φ$

Andere stix macros

$∰_{A} f (x) d x$
$∯_{A} f (x) d x$
$∳_{A} f (x) d x$
$∳_{A} f (x) d x$
$∲_{A} f (x) d x$
$∲_{A} f (x) d x$

Summen

MathML

$T^{(n)} (V) : = ⨁_{i = 0}^{n} V^{\otimes i}$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$F_{N} (s) = \prod_{\binom{p \leq N}{p Primzahl}} \sum_{k = 0}^{\infty} \frac{f (p^{k})}{p^{k s}}$
$F_{N} (s) = \prod_{\binom{p \leq N}{p Primzahl}} \sum_{k = 0}^{\infty} \frac{f (p^{k})}{p^{k s}}$
$\sum_{I, J} f_{I, J} d z_{I} \land d {\overline{z}}_{J} : = \sum_{\overset{1 \leq i_{1} < \dots < i_{p} \leq p}{1 \leq j_{1} < \dots < j_{q} \leq q}}^{} f_{i_{1}, \dots i_{p}, j_{1}, \dots j_{q}} d z_{i_{1}} \land \dots \land d z_{i_{p}} \land d {\overline{z}}_{j_{1}} \land \dots \land d {\overline{z}}_{j_{q}}$

Native

$T^{(n)} (V) : = ⨁_{i = 0}^{n} V^{\otimes i}$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$F_{N} (s) = \prod_{\binom{p \leq N}{p Primzahl}} \sum_{k = 0}^{\infty} \frac{f (p^{k})}{p^{k s}}$
$F_{N} (s) = \prod_{\binom{p \leq N}{p Primzahl}} \sum_{k = 0}^{\infty} \frac{f (p^{k})}{p^{k s}}$
$\sum_{I, J} f_{I, J} d z_{I} \land d {\overline{z}}_{J} : = \sum_{\overset{1 \leq i_{1} < \dots < i_{p} \leq p}{1 \leq j_{1} < \dots < j_{q} \leq q}}^{} f_{i_{1}, \dots i_{p}, j_{1}, \dots j_{q}} d z_{i_{1}} \land \dots \land d z_{i_{p}} \land d {\overline{z}}_{j_{1}} \land \dots \land d {\overline{z}}_{j_{q}}$

Klammern

MathML

$γ x + \ln [Γ (x + 1)] = \sum_{n = 1}^{\infty} [\frac{x}{n} - \ln (1 + \frac{x}{n})]$

Native

$γ x + \ln [Γ (x + 1)] = \sum_{n = 1}^{\infty} [\frac{x}{n} - \ln (1 + \frac{x}{n})]$

Weiteres

MathML

$Hom$
$\overline{\partial} f : = \sum_{j = 1}^{n} \frac{\partial}{\partial {\overline{z}}_{j}} f d {\overline{z}}_{j}$
$\underset{i \in I}{⋁} 𝒜_{i} = σ (⋃_{i \in I} 𝒜_{i})$
$⋂_{n \in ℕ} A_{n} = (⋃_{n \in ℕ} A_{n}^{c})^{c}$
${\begin{matrix} \frac{1}{2 π} \ln | x |, & n = 2, \\ - \frac{1}{(n - 2) ω_{n}} \frac{1}{| x |^{n - 2}}, & n > 2 \end{matrix}$

Native

$Hom$
$0 ⟶ Γ^{\infty} (E_{0}) \overset{D_{0}}{⟶} Γ^{\infty} (E_{1}) \overset{D_{1}}{⟶} \dots \overset{D_{m - 1}}{⟶} Γ^{\infty} (E_{m}) ⟶ 0$
$\overline{\partial} f : = \sum_{j = 1}^{n} \frac{\partial}{\partial {\overline{z}}_{j}} f d {\overline{z}}_{j}$
$\underset{i \in I}{⋁} 𝒜_{i} = σ (⋃_{i \in I} 𝒜_{i})$
$⋂_{n \in ℕ} A_{n} = (⋃_{n \in ℕ} A_{n}^{c})^{c}$
${\begin{matrix} \frac{1}{2 π} \ln | x |, & n = 2, \\ - \frac{1}{(n - 2) ω_{n}} \frac{1}{| x |^{n - 2}}, & n > 2 \end{matrix}$

Buchstaben

MathML

$ℕ, ℤ, ℚ, ℝ, ℝ, ℂ, ℂ, ℍ, 𝕒$
$𝒜 ℬ 𝒞 𝒟 ℰ ℱ 𝒢 ℋ ℐ 𝒥 𝒦 ℒ ℳ 𝒩$

Native

$ℕ, ℤ, ℚ, ℝ, ℝ, ℂ, ℂ, ℍ, 𝕒$
$𝒜 ℬ 𝒞 𝒟 ℰ ℱ 𝒢 ℋ ℐ 𝒥 𝒦 ℒ ℳ 𝒩$

Matrizen

MathMl

$A = (\begin{matrix} 3 & 2 & 1 \\ 1 & 0 & 2 \end{matrix}) \in ℝ^{2 \times 3}$

Native

$A = (\begin{matrix} 3 & 2 & 1 \\ 1 & 0 & 2 \end{matrix}) \in ℝ^{2 \times 3}$

Fließtext (Satz von Atkinson)

MathMl

Nach dem Satz von Atkinson ist ein Operator $A : X \to Y$ genau dann ein Fredholm-Operator, wenn es Operatoren $B_{1}, B_{2}$ und kompakte Operatoren $K_{1}, K_{2}$ gibt, so dass $A B_{1} = I_{Y} - K_{1}$ und $B_{2} A = I_{X} - K_{2}$ gilt, das heißt wenn $A$ modulo kompakter Operatoren invertierbar ist. Insbesondere ist ein beschränkter Operator $A : X \to X$ genau dann ein Fredholm-Operator, wenn seine Klasse $[A]_{𝒞 (X)}$ in der Calkin-Algebra $ℬ (X) / 𝒞 (X)$ invertierbar ist.

Native

Nach dem Satz von Atkinson ist ein Operator $A : X \to Y$ genau dann ein Fredholm-Operator, wenn es Operatoren $B_{1}, B_{2}$ und kompakte Operatoren $K_{1}, K_{2}$ gibt, so dass $A B_{1} = I_{Y} - K_{1}$ und $B_{2} A = I_{X} - K_{2}$ gilt, das heißt wenn $A$ modulo kompakter Operatoren invertierbar ist. Insbesondere ist ein beschränkter Operator $A : X \to X$ genau dann ein Fredholm-Operator, wenn seine Klasse $[A]_{𝒞 (X)}$ in der Calkin-Algebra $ℬ (X) / 𝒞 (X)$ invertierbar ist.

Algin

MathMl

$\begin{aligned} (\frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}}) \cdot (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & \frac{\partial^{2}}{\partial r^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{2}{r} \frac{\partial}{\partial r} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} \frac{\partial}{\partial θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial}{\partial φ} (v_{r, φ} {\hat{e}}_{r} + \sin θ v_{r} {\hat{e}}_{φ} + v_{θ, φ} {\hat{e}}_{θ} + \cos θ v_{θ} {\hat{e}}_{φ} + v_{φ, φ} {\hat{e}}_{φ} - \sin θ v_{φ} {\hat{e}}_{r} - \cos θ v_{φ} {\hat{e}}_{θ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} (v_{r, θ θ} {\hat{e}}_{r} + v_{r, θ} {\hat{e}}_{θ} + v_{r, θ} {\hat{e}}_{θ} - v_{r} {\hat{e}}_{r} + v_{θ, θ θ} {\hat{e}}_{θ} - v_{θ, θ} {\hat{e}}_{r} - v_{θ, θ} {\hat{e}}_{r} - v_{θ} {\hat{e}}_{θ} + v_{φ, θ θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} (v_{r, φ φ} {\hat{e}}_{r} + \sin θ v_{r, φ} {\hat{e}}_{φ} + \sin θ v_{r, φ} {\hat{e}}_{φ} - \sin^{2} θ v_{r} {\hat{e}}_{r} - \sin θ \cos θ v_{r} {\hat{e}}_{θ} \\ + v_{θ, φ φ} {\hat{e}}_{θ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} - \sin θ \cos θ v_{θ} {\hat{e}}_{r} - \cos^{2} θ v_{θ} {\hat{e}}_{θ} \\ + v_{φ, φ φ} {\hat{e}}_{φ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \sin^{2} θ v_{φ} {\hat{e}}_{φ} \\ - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \cos^{2} θ v_{φ} {\hat{e}}_{φ}) \\ = & (v_{r, r r} + \frac{2}{r} v_{r, r} + \frac{1}{r^{2}} v_{r, θ θ} + \frac{1}{r^{2} \tan θ} v_{r, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{r, φ φ} \\ - \frac{1}{r^{2}} v_{r} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2} \tan θ} v_{θ} - \frac{1}{r^{2}} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{θ} - \frac{1}{r^{2} \sin θ} v_{φ, φ} - \frac{1}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (v_{θ, r r} + \frac{2}{r} v_{θ, r} + \frac{1}{r^{2}} v_{θ, θ θ} + \frac{1}{r^{2} \tan θ} v_{θ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{θ, φ φ} \\ + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2}} v_{θ} + \frac{1}{r^{2} \tan θ} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{r} - \frac{\cos^{2} θ}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (v_{φ, r r} + \frac{2}{r} v_{φ, r} + \frac{1}{r^{2}} v_{φ, θ θ} + \frac{1}{r^{2} \tan θ} v_{φ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{φ, φ φ} \\ + \frac{2}{r^{2} \sin θ} v_{r, φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{\sin^{2} θ + \cos^{2} θ}{r^{2} \sin^{2} θ} v_{φ}) {\hat{e}}_{φ} \\ = & (Δ v_{r} - \frac{2}{r^{2}} v_{r} - \frac{2}{r^{2}} v_{θ, θ} - \frac{2}{r^{2} \tan θ} v_{θ} - \frac{2}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (Δ v_{θ} + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (Δ v_{φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{1}{r^{2} \sin^{2} θ} v_{φ} + \frac{2}{r^{2} \sin θ} v_{r, φ}) {\hat{e}}_{φ} \end{aligned}$

Native

$\begin{aligned} (\frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}}) \cdot (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & \frac{\partial^{2}}{\partial r^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{2}{r} \frac{\partial}{\partial r} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} \frac{\partial}{\partial θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial}{\partial φ} (v_{r, φ} {\hat{e}}_{r} + \sin θ v_{r} {\hat{e}}_{φ} + v_{θ, φ} {\hat{e}}_{θ} + \cos θ v_{θ} {\hat{e}}_{φ} + v_{φ, φ} {\hat{e}}_{φ} - \sin θ v_{φ} {\hat{e}}_{r} - \cos θ v_{φ} {\hat{e}}_{θ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} (v_{r, θ θ} {\hat{e}}_{r} + v_{r, θ} {\hat{e}}_{θ} + v_{r, θ} {\hat{e}}_{θ} - v_{r} {\hat{e}}_{r} + v_{θ, θ θ} {\hat{e}}_{θ} - v_{θ, θ} {\hat{e}}_{r} - v_{θ, θ} {\hat{e}}_{r} - v_{θ} {\hat{e}}_{θ} + v_{φ, θ θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} (v_{r, φ φ} {\hat{e}}_{r} + \sin θ v_{r, φ} {\hat{e}}_{φ} + \sin θ v_{r, φ} {\hat{e}}_{φ} - \sin^{2} θ v_{r} {\hat{e}}_{r} - \sin θ \cos θ v_{r} {\hat{e}}_{θ} \\ + v_{θ, φ φ} {\hat{e}}_{θ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} - \sin θ \cos θ v_{θ} {\hat{e}}_{r} - \cos^{2} θ v_{θ} {\hat{e}}_{θ} \\ + v_{φ, φ φ} {\hat{e}}_{φ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \sin^{2} θ v_{φ} {\hat{e}}_{φ} \\ - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \cos^{2} θ v_{φ} {\hat{e}}_{φ}) \\ = & (v_{r, r r} + \frac{2}{r} v_{r, r} + \frac{1}{r^{2}} v_{r, θ θ} + \frac{1}{r^{2} \tan θ} v_{r, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{r, φ φ} \\ - \frac{1}{r^{2}} v_{r} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2} \tan θ} v_{θ} - \frac{1}{r^{2}} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{θ} - \frac{1}{r^{2} \sin θ} v_{φ, φ} - \frac{1}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (v_{θ, r r} + \frac{2}{r} v_{θ, r} + \frac{1}{r^{2}} v_{θ, θ θ} + \frac{1}{r^{2} \tan θ} v_{θ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{θ, φ φ} \\ + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2}} v_{θ} + \frac{1}{r^{2} \tan θ} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{r} - \frac{\cos^{2} θ}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (v_{φ, r r} + \frac{2}{r} v_{φ, r} + \frac{1}{r^{2}} v_{φ, θ θ} + \frac{1}{r^{2} \tan θ} v_{φ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{φ, φ φ} \\ + \frac{2}{r^{2} \sin θ} v_{r, φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{\sin^{2} θ + \cos^{2} θ}{r^{2} \sin^{2} θ} v_{φ}) {\hat{e}}_{φ} \\ = & (Δ v_{r} - \frac{2}{r^{2}} v_{r} - \frac{2}{r^{2}} v_{θ, θ} - \frac{2}{r^{2} \tan θ} v_{θ} - \frac{2}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (Δ v_{θ} + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (Δ v_{φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{1}{r^{2} \sin^{2} θ} v_{φ} + \frac{2}{r^{2} \sin θ} v_{r, φ}) {\hat{e}}_{φ} \end{aligned}$

Intbar

Inhaltsverzeichnis

Integrale

MathML

Mittelwertintegrale

Normale Integrale

Native

Mittelwertintegrale

Normale Integrale

Andere stix macros

Summen

MathML

Native

Klammern

MathML

Native

Weiteres

MathML

Native

Buchstaben

MathML

Native

Matrizen

MathMl

Native

Fließtext (Satz von Atkinson)

MathMl

Native

Algin

MathMl

Native

Widetilde & Overline

MathMl

Native

Potenzen & Indizes

MathMl

Native